seasonal MK test different from standalone MK test for each month?

[Louisa121]

Question:
When I run the example in the doc:
n_season = 4

y0 = [datetime(2000, 1, 1), datetime(2000, 4, 1), datetime(2000, 7, 1), datetime(2000, 10, 1)]
multi_obs_dts = [np.array([y+timedelta(days=365*item) for item in range(19)]) for y in y0]
multi_obs = [np.array([3.20, 2.92, 3.95, 1.80, 2.45, 2.70, 2.22, 2.10, 2.08, 2.21, 1.93, 2.15, 2.03, 1.82, 1.94, 2.24, 1.67, 1.34, 1.61]),
np.array([3.92, 2.99, 4.37, 3.04, 3.12, 4.07, 3.91, 3.42, 2.94, 3.14, 2.53, 2.80, 2.98, 2.86, 3.22, 2.31, 2.03, 1.59, 2.14]),
np.array([4.56, 4.13, 4.31, 1.83, 3.22, 5.06, 4.39, 4.13, 4.06, 3.20, 4.01, 3.62, 3.78, 3.61, 3.42, 3.65, 2.39, 3.01, 3.03]),
np.array([4.22, 4.78, 2.96, 3.23, 2.82, 2.96, 3.12, 3.49, 2.73, 2.61, 3.00, 2.66, 3.49, 2.58, 2.32, 2.10, 2.38, 2.29, 2.07])]

out = mk.mk_temp_aggr(multi_obs_dts, multi_obs, 0.001)

for n in range(n_season):
print('Season {ind}:'.format(ind=n+1), out[n])

print('Combined yearly trend:', out[n_season])

I get the same result as the doc provided. However, if I run them separately, e.g. mk.mk_temp_aggr(multi_obs_dts[:2], multi_obs[:2], 0.001), the results are different. Why could it be like this? Shouldn't the first four results in out be the same as the standalone MK test for each season?

[mcollaudcoen]

Hi Louisa,

Yes the results can be different since the autocorrelation (and its correction) is applied on the whole time series to ensure that is it similar for all seasons, what leads to the best results. If you apply the test on each season, the autocorrelation is computed for each season separately. i tested that this solution can lead to larger discrepancy than if the same autocorrelation correction if applied to the whole time series.
Best
Martine

[Louisa121]

Thank you very much for your prompt response! Does it mean that it makes more sense to use the results from the seasonal MK for each month?
Besides, how is the annual trend determined?
For example, below is the result of the 12 months. I feel it's weird that in the combined result, it shows ss of 90, but the slope is nan. Does it make sense? What could be the issue? Thank you!
{0: {'p': 0.15085154251250898,
'ss': -1,
'slope': -1.3067294431631238,
'ucl': 0.2761673252480693,
'lcl': -2.9226746298788897},
1: {'p': 0.8338312543415771,
'ss': 0,
'slope': 0.6135143512552871,
'ucl': 2.903093465338299,
'lcl': -1.1082519252121574},
2: {'p': 0.39244216795297104,
'ss': 0,
'slope': -0.547318366463335,
'ucl': 0.7426233627031255,
'lcl': -1.7385120765260151},
3: {'p': 0.0003102507580297065,
'ss': 90,
'slope': -2.1189682289250236,
'ucl': -1.3069764407489053,
'lcl': -3.0664384837728194},
4: {'p': 0.06041394837486713,
'ss': 90,
'slope': -0.5524386767989283,
'ucl': -0.12866493649697305,
'lcl': -0.9930350576019779},
5: {'p': 0.668585226133676,
'ss': -1,
'slope': -0.0023604460825425266,
'ucl': 0.24932489771864147,
'lcl': -0.32330898054376056},
6: {'p': 0.005315307586087004,
'ss': 90,
'slope': -0.9468708109955086,
'ucl': -0.47696650438404237,
'lcl': -1.4003918768236217},
7: {'p': 0.15366343214750433,
'ss': -1,
'slope': -0.5154798129481447,
'ucl': 0.1513983804314962,
'lcl': -1.0847463102630563},
8: {'p': 0.12196121661564518,
'ss': -1,
'slope': -0.2743810505910557,
'ucl': 0.13939161683922824,
'lcl': -0.6659592828283087},
9: {'p': 0.002786212311651992,
'ss': 90,
'slope': -0.5321329286865193,
'ucl': -0.19808841011723888,
'lcl': -0.9036204866193961},
10: {'p': 0.008748274312619486,
'ss': 90,
'slope': -0.6800596124905317,
'ucl': -0.18719555892409512,
'lcl': -1.2407630225328654},
11: {'p': 0.19549769192437694,
'ss': -1,
'slope': -1.055013921257681,
'ucl': 0.27872546972066564,
'lcl': -2.0663837612519598},
12: {'p': 7.950771152565039e-10,
'ss': 90,
'slope': nan,
'ucl': nan,
'lcl': nan}}

[mcollaudcoen]

Using seasons or months depends on your time series and research questions. E.g. if the seasons are not well defined or have not the same lengths, it's better to use months.
the yearly trend is the median of all seasonal or monthly trend.

[Louisa121]

Thank you. If I would like to know the Sen's slope for each month, is it recommended to do it one month at a time? Or is it better to input a list of 12 months into the seasonal MK test at one time and use that result for each month?

[mcollaudcoen]

It's better to use the seasonal MK test at one time and use then the results for each month.